Understanding nonlinearity in statistical image reconstruction for nuclear medicine

This study aimed to propose a definition of linearity in image reconstruction and demonstrate, by reductio ad absurdum, that the row-action maximum likelihood algorithm (RAMLA) and ordered subset expectation maximization (OSEM) are nonlinear when the number of iterations is low and linear approximation when the number of iterations increases. Block sequential regularized expectation maximization (BSREM) and one-step late maximum a posteriori expectation maximization (OSLEM), which serve as regularized versions of RAMLA and OSEM, respectively, remain nonlinear regardless of the number of iterations. Simulations using ideal two-dimensional (2D) parallel beam projections validated the results of the reductio ad absurdum proof. The three numerical phantoms were point source ${\overline{x}}_1$ , represented by 2D Gaussian with a full width at half maximum of 3 pixels positioned at the center of disk background; point source ${\overline{x}}_2$ , separated by 24 pixels along the x-axis; and point source ${\overline{x}}_3$ , is the sum of ${\overline{x}}_1$ and ${\overline{x}}_2$ . In numerical experiment, when the difference of the area under the curve (AUC) or recovery for reconstructed image of ${\overline{x}}_3$ and the summed reconstructed images of ${\overline{x}}_1$ and ${\overline{x}}_2$ is within reference values, or when AUC profiles are visually consistent, we defined image reconstruction as linear approximation. RAMLA and OSEM were deemed nonlinear when less than 20 iterations were performed with 64 subsets and linear approximation when $\ge 20$ iterations were used. By contrast, BSREM and OSLEM remained nonlinear. Algebraic reconstruction technique is linear and its regularized variant has a tendency of linear approximation, indicating that the same regularization function works differently in linear and nonlinear image reconstructions.